Exercise 10.1.1 Consider the following functions T: R³ → R². For each of these functions T, show that it is not linear either by showing that T does not preserve addition, or by showing that it does not preserve scalar multiplication, or by showing that it does not preserve the zero vector. (a) T (b) T (c) T Z X 3-4 = Z X Z x+2y+3z +1 2y-s = 2y+3x+z 1. sinx+2y+3z 2y+3x+z
Exercise 10.1.1 Consider the following functions T: R³ → R². For each of these functions T, show that it is not linear either by showing that T does not preserve addition, or by showing that it does not preserve scalar multiplication, or by showing that it does not preserve the zero vector. (a) T (b) T (c) T Z X 3-4 = Z X Z x+2y+3z +1 2y-s = 2y+3x+z 1. sinx+2y+3z 2y+3x+z
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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can you help me solve part b and c please?
![**Exercise 10.1.1**
Consider the following functions \( T : \mathbb{R}^3 \to \mathbb{R}^2 \). For each of these functions \( T \), show that it is not linear either by showing that \( T \) does not preserve addition, or by showing that it does not preserve scalar multiplication, or by showing that it does not preserve the zero vector.
(a) \[
T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y + 3z + 1 \\ 2y - 3x + z \end{bmatrix}
\]
(b) \[
T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y^2 + 3z \\ 2y + 3x + z \end{bmatrix}
\]
(c) \[
T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \sin x + 2y + 3z \\ 2y + 3x + z \end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52fb4760-8b56-41d1-811f-26cd2f9d5b41%2Fd2ddfcd3-8f9e-4768-9d72-e558b56db821%2Fys80y09_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 10.1.1**
Consider the following functions \( T : \mathbb{R}^3 \to \mathbb{R}^2 \). For each of these functions \( T \), show that it is not linear either by showing that \( T \) does not preserve addition, or by showing that it does not preserve scalar multiplication, or by showing that it does not preserve the zero vector.
(a) \[
T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y + 3z + 1 \\ 2y - 3x + z \end{bmatrix}
\]
(b) \[
T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y^2 + 3z \\ 2y + 3x + z \end{bmatrix}
\]
(c) \[
T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \sin x + 2y + 3z \\ 2y + 3x + z \end{bmatrix}
\]
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